The minimum by default fields required to add a customer are:
1. Form of contact
2. First Name
3. Last Name
4. At least one phone number
TIP: If you do not have your client's phone or email, you can enter your business's email or phone number.
You can set other fields as required from: Setup -> Required Fields
IMPORTANT!! If you customize other required fields, you MUST complete all fields when adding a client. If you are adding a client and you notice that the information is not being saved or if it is asking you for other fields it means you have customized the "Required Fields" section.